I have a fiber bunde $F\to E\stackrel \pi \to B$ and i want to calculate its first Chern class $c_1(E)=c_1(TE)$. How can i do this?
I read here, that $$TE \stackrel \sim = \pi^* TB \oplus T_\pi E, $$ where $T_\pi E$ consists of the tangent vectors tangent to the fibers.
So $$c_1(TE)= \pi^* c_1(TB) + c_1(T_\pi E).$$
Its clear how to calculate $\pi^* c_1(TB)$, but what about $c_1(T_\pi E)$?
Is it true that $c_1(T_\pi E)= c_1(TF)$?. The bundle $T_\pi E$ seems to be in some sense $B\times TF$, but i don't know how to make this precise.